Modern warfare increasingly relies on drone technology, where Unmanned Aerial Vehicles (UAVs) outperform manned aircraft in cost-effectiveness, deployment flexibility, and survivability. However, single-UAV systems face functional limitations in complex missions. Multi-UAV cooperative formations enhance capabilities through distributed sensing and coordinated strikes. High-precision formation tracking ensures synchronized mission execution while maintaining compact formations reduces detectability. Traditional methods neglect temporal constraints and require extensive parameter tuning for precision. Integrating fixed-time convergence with prescribed performance control (PPC) addresses these gaps by guaranteeing formation accuracy within user-defined timeframes.
System Modeling and Preliminaries
Consider a multi-Unmanned Aerial Vehicle system with \(N\) followers and one virtual leader. Each quadrotor’s dynamics are decoupled into position and attitude subsystems:
$$ \begin{align*}
\dot{\mathbf{p}}_i &= \mathbf{v}_i \\
\dot{\mathbf{v}}_i &= \mathbf{f}_{pi} + \mathbf{b}_{pi}\mathbf{u}_{pi} \\
\dot{\boldsymbol{\delta}}_i &= \boldsymbol{\omega}_i \\
\dot{\boldsymbol{\omega}}_i &= \mathbf{f}_{\delta i} + \mathbf{b}_{\delta i}\mathbf{u}_{\delta i}
\end{align*} $$
where \(\mathbf{p}_i = [x_i, y_i, z_i]^T\), \(\mathbf{v}_i\) = linear velocities, \(\boldsymbol{\delta}_i = [\phi_i, \theta_i, \psi_i]^T\) = Euler angles, and \(\boldsymbol{\omega}_i\) = angular velocities. Control inputs \(\mathbf{u}_{pi} = [u_{x,i}, u_{y,i}, u_{z,i}]^T\) and \(\mathbf{u}_{\delta i} = [u_{\phi,i}, u_{\theta,i}, u_{\psi,i}]^T\) govern position and attitude, respectively.
Prescribed Performance Control Framework
PPC ensures tracking error \(e(t)\) evolves within designer-specified bounds:
$$ -\rho(t) < e(t) < \rho(t) $$
where \(\rho(t)\) is a performance function. We design a fixed-time convergent \(\rho(t)\):
$$ \rho(t) = \begin{cases}
(\rho_0 – \rho_{\infty})(1 – t/T_f)^\tau + \rho_{\infty}, & t \leq T_f \\
\rho_{\infty}, & t > T_f
\end{cases} $$
with \(\tau \in (0.5,1)\), \(\rho_0 > \rho_{\infty} > 0\), and \(T_f\) as the user-specified convergence time. A tangent-type error transformation facilitates unconstrained controller design:
$$ \epsilon(t) = \tan\left(\frac{\pi e(t)}{2\rho(t)}\right) $$
Formation Tracking Errors
Global formation errors for position and attitude are:
$$ \begin{align*}
\mathbf{e}_{pi} &= \sum_{j \in \mathcal{N}_i} a_{ij}(\mathbf{p}_i – \mathbf{p}_j – \Delta_{ij}) + b_i(\mathbf{p}_i – \mathbf{y}_r – \Delta_i) \\
\mathbf{e}_{\delta i} &= \sum_{j \in \mathcal{N}_i} a_{ij}(\boldsymbol{\delta}_i – \boldsymbol{\delta}_j) + b_i(\boldsymbol{\delta}_i – \boldsymbol{\delta}_{d,i})
\end{align*} $$
where \(\Delta_i\) = desired relative position to leader, \(\mathbf{y}_r\) = leader’s trajectory, and \(a_{ij}\), \(b_i\) = adjacency weights.
Dual-Loop Control Design
Outer Loop: Fixed-Time PPC Formation Control
Define auxiliary variables \(\mathbf{q}_i = \mathbf{e}_{pi} + k_v \mathbf{e}_{vi}\) with \(k_v > 0\). The FTPPC controller ensures \(\mathbf{q}_i\) converges within \(\pm\rho(t)\):
$$ \mathbf{u}_{pi} = \frac{1}{\sum_{j} a_{ij} + b_i} \left[ \sum_{j \in \mathcal{N}_i} a_{ij} \mathbf{u}_{pj} – k_1 \boldsymbol{\eta}_i^{-1} \boldsymbol{\epsilon}_{qi} – \mathbf{Y}_i \right] $$
where \(\boldsymbol{\epsilon}_{qi} = \tan(\pi \mathbf{q}_i / (2\rho))\), \(\boldsymbol{\eta}_i = \text{diag}(\partial \epsilon_{qi,k}/\partial q_{i,k})\), and \(\mathbf{Y}_i\) aggregates neighbor information.
Inner Loop: Attitude Control
A proportional controller achieves rapid attitude synchronization:
$$ \mathbf{u}_{\delta i} = \frac{1}{\sum_{j} a_{ij} + b_i} \left[ \sum_{j \in \mathcal{N}_i} a_{ij} \mathbf{u}_{\delta j} – k_a \mathbf{q}_{ai} – \mathbf{Y}_{ai} \right] $$
where \(\mathbf{q}_{ai} = k_{\omega} \mathbf{e}_{\delta i} + k_{\delta} \mathbf{e}_{\omega i}\). Reference roll/pitch angles derive from position control inputs:
$$ \begin{align*}
\phi_{d,i} &= \arcsin\left( \frac{-u_{x,i}\sin\psi_i + u_{y,i}\cos\psi_i}{\sqrt{u_{x,i}^2 + u_{y,i}^2 + u_{z,i}^2}} \right) \\
\theta_{d,i} &= \arcsin\left( \frac{u_{x,i}\cos\psi_i + u_{y,i}\sin\psi_i}{\cos\phi_{d,i} \sqrt{u_{x,i}^2 + u_{y,i}^2 + u_{z,i}^2}} \right)
\end{align*} $$
Stability Analysis
Consider the Lyapunov function \(V = \frac{1}{2} \boldsymbol{\epsilon}_{qi}^T \boldsymbol{\epsilon}_{qi} + \frac{1}{2} \mathbf{q}_{ai}^T \mathbf{q}_{ai}\). Its derivative satisfies:
$$ \dot{V} = -k_1 \boldsymbol{\epsilon}_{qi}^T \boldsymbol{\eta}_i \mathbf{q}_i – k_a \mathbf{q}_{ai}^T \mathbf{q}_{ai} < 0 $$
confirming global asymptotic stability. The fixed-time performance function guarantees convergence within \(T_f\).
Simulation Results
A hexagonal formation of six UAVs tracking a virtual leader validates the approach. Communication topology and parameters are:
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| \(m\) | 2.0 kg | \(I_{xx}\) | 0.55 kg·m² |
| \(g\) | 9.8 m/s² | \(I_{yy}\) | 0.51 kg·m² |
| \(l\) | 0.1 m | \(I_{zz}\) | 0.96 kg·m² |
| \(J_p\) | 0.01 m | \(\rho_0/\rho_{\infty}\) | 6/0.1 |
| \(T_f\) | 35 s | \(\tau\) | 0.55 |
Key observations from simulations:
- Global position errors converge below \(\rho_{\infty} = 0.1\) m within \(T_f = 35\) s
- Attitude synchronization completes in under 5 s
- Inter-agent distances stabilize at desired 30 m separation
- Control inputs remain smooth without chattering
Comparative advantages of this drone technology approach:
$$ \begin{array}{|l|c|c|}
\hline
\text{Criterion} & \text{Conventional SMC} & \text{Proposed FTPPC} \\
\hline
\text{Convergence Time} & \text{Asymptotic} & \text{Fixed-Time } (T_f) \\
\hline
\text{Steady-State Error} & >0.5 \text{ m} & <0.1 \text{ m} \\
\hline
\text{Parameter Tuning} & \text{Extensive} & \text{Minimal} \\
\hline
\text{Input Chattering} & \text{Present} & \text{Absent} \\
\hline
\end{array} $$
Conclusion
This work presents a fixed-time prescribed performance control for multi-Unmanned Aerial Vehicle formations. The dual-loop architecture achieves:
- Precise formation tracking with steady-state error \(< \rho_{\infty}\)
- Guaranteed convergence within user-defined time \(T_f\)
- Distributed implementation using neighbor-only information
Future research will integrate collision avoidance and connectivity maintenance under communication constraints. Robustness against wind disturbances and actuator faults will enhance practical applicability of drone technology in military and civilian operations. The prescribed performance framework offers significant potential for time-critical multi-UAV missions requiring high coordination accuracy.